arXiv research

The paper studies symplectic structures on character varieties of Sasakian threefolds.

We prove that every homomorphism from the elementary Chevalley group over a finitely generated unital commutative ring associated with reduced irreducible classical root system of rank at least 2, and ME analogues of such groups, into acylindrically hyperbolic groups has an absolutely elliptic image. This result provid…

Any endomorphism of a finitely generated free group naturally descends to an injective endomorphism of its stable quotient. In this paper, we prove a geometric incarnation of this phenomenon: namely, that every expanding irreducible train track map inducing an endomorphism of the fundamental group gives rise to an expa…

If $G$ is a semisimple Lie group of real rank at least 2 and $Γ$ is an irreducible lattice in $G$, then every homomorphism from $Γ$ to the outer automorphism group of a finitely generated free group has finite image.

The study classifies and characterizes totally symmetric sets in the general linear group.

We investigate the flat holomorphic vector bundles over compact complex parallelizable manifolds $G / Γ$, where $G$ is a complex connected Lie group and $Γ$ is a cocompact lattice in it. The main result proved here is a structure theorem for flat holomorphic vector bundles $E_ρ$ associated to any irreducible representa…

This is the third article in the series begun with [BonWon3, BonWon4], devoted to finite-dimensional representations of the Kauffman bracket skein algebra of an oriented surface $S$. In [BonWon3] we associated a classical shadow to an irreducible representation $ρ$ of the skein algebra, which is a character $r_ρ\in \ma…

We prove that instanton L-space knots are fibered and strongly quasipositive. Our proof differs conceptually from proofs of the analogous result in Heegaard Floer homology, and includes a new decomposition theorem for cobordism maps in framed instanton Floer homology akin to the $\textrm{Spin}^c$ decompositions of cobo…

In the present paper, we study the Sp-module structure of the cokernel of the Johnson homomorphism of the mapping class groups of surfaces. We detect the Sp-irreducible components with highest weight [1^k] (and [k]) in the cokernel. We also show that the multiplicities of them is equal to one.

We present a complete classification and the construction of $\mathrm{Mp}(2n+2,\mathbb{R})$-equivariant differential operators acting on the principal series representations, associated to the contact projective geometry on $\mathbb{RP}^{2n+1}$ and induced from the irreducible $\mathrm{Mp}(2n,\mathbb{R})$-submodules of…

We study finite-dimensional representations of the Kauffman skein algebra of a surface S. In particular, we construct invariants of such irreducible representations when the underlying parameter q is a root of unity. The main one of these invariants is a point in the character variety consisting of group homomorphisms …

Let M^3 be a compact, oriented, irreducible, and boundary incompressible 3-manifold. Assume that its fundamental group is without rank two abelian subgroups and its boundary is non-empty. We will show that every homomorphism from pi_1(M) to PSL(2,C) which is not `boundary elementary' is induced by a possibly branched c…

Let $φ\in \mbox{Out}(F_n)$ be a free group outer automorphism that can be represented by an expanding, irreducible train-track map. The automorphism $φ$ determines a free-by-cyclic group $Γ=F_n \rtimes_φ\mathbb Z,$ and a homomorphism $α\in H^1(Γ; \mathbb Z)$. By work of Neumann, Bieri-Neumann-Strebel and Dowdall-Kapovi…

A totally geodesic map $f:\mathcal X_1\to\mathcal X_2$ between Hermitian symmetric spaces is tight if its image contains geodesic triangles of maximal area. Tight maps were first introduced in [BIW09], and were classified in [Ham13, Ham14, HO14] in the case of irreducible domain. We complete the classification by analy…

The paper connects two skein algebras and characterizes their representations.

Suppose M is a connected, open, orientable, irreducible 3-manifold which is not homeomorphic to R^3. Given a compact 3-manifold J in M which satisfies certain conditions, Brin and Thickstun have associated to it an open neighborhood V$ called an end reduction of M at J. It has some useful properties which allow one to …

Study reveals uniform difference in stretch factors between genus two handlebody group and outer automorphism group.

Study of low dimensional representations of mapping class groups of surfaces, focusing on genus ≥ 7.

Classifies invariant differential operators on a specific geometric space.

Let $K\backslash G$ be an irreducible Hermitian symmetric space of noncompact type and $Γ\,\subset\, G$ a closed torsionfree discrete subgroup. Let $X$ be a compact Kähler manifold and $ρ\, :\, π_1(X, x_0)\,\longrightarrow\, Γ$ a homomorphism such that the adjoint action of $ρ(π_1(X, x_0))$ on $\text{Lie}(G)$ is comple…

We investigate the representation theory of the polynomial core of the quantum Teichmuller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of S. Our main result is that irreducible finite-dimensional represent…

A Kauffman bracket on a surface is an invariant for framed links in the thickened surface, satisfying the Kauffman skein relation and multiplicative under superposition. This includes representations of the skein algebra of the surface. We show how an irreducible representation of the skein algebra usually specifies a …

Proves a theorem about groups and 3-manifolds.

We recently defined invariants of contact 3-manifolds using a version of instanton Floer homology for sutured manifolds. In this paper, we prove that if several contact structures on a 3-manifold are induced by Stein structures on a single 4-manifold with distinct Chern classes modulo torsion then their contact invaria…

Characterizes a general range decreasing group homomorphism.

Given a free-by-cyclic group $G = F_N \rtimes_\varphi \mathbb{Z}$ determined by any outer automorphism $\varphi \in \mathrm{Out}(F_N)$ which is represented by an expanding irreducible train-track map $f$, we construct a $K(G,1)$ $2$-complex $X$ called the folded mapping torus of $f$, and equip it with a semiflow. We sh…

Uniformly random permutations converge to regular representation on surface groups.

We present a construction that produces infinite classes of Kähler groups that arise as fundamental groups of fibres of maps to higher dimensional tori. Following the work of Delzant and Gromov, there is great interest in knowing which subgroups of direct products of surface groups are Kähler. We apply our construction…

We extend certain homomorphisms defined on the higher Torelli subgroups of the mapping class group to crossed homomorphisms defined on the entire mapping class group. In particular, for every $k\geq 2$, we construct a crossed homomorphism $ε_k$ which extends Morita's homomorphism $\tilde τ_k$ to the entire mapping clas…

Maps preserving Carathéodory distance between symmetric domains are rigid.

Two crossing homomorphisms on braid groups are shown to be equivalent.

The study classifies homomorphisms from mapping class groups using finite subgroups.

Study of Chebyshev-Frobenius homomorphism in 3-manifold skein modules.

Graph homomorphism numbers embed graphs for classification.

New conditions for weighted composition operators in group homomorphisms.

Maps between certain configuration spaces are rigid and affine equivalent.

We study the abelianization of Kontsevich's Lie algebra associated with the Lie operad and some related problems. Calculating the abelianization is a long-standing unsolved problem, which is important in at least two different contexts: constructing cohomology classes in $H^k(\mathrm{Out}(F_r);\mathbb Q)$ and related g…

Study homomorphisms from groups to 3-manifold fundamental groups.

New homomorphism from Khovanov homology for knot concordance.

In this paper we use character variety methods to study homomorphisms between the fundamental groups of 3-manifolds, in particular those induced by non-zero degree maps. A {\it knot manifold} is a compact, connected, irreducible, orientable 3-manifold whose boundary is an incompressible torus. A {\it virtual epimorphis…

We introduce the notion of tight homomorphism into a locally compact group with nonvanishing bounded cohomology and study these homomorphisms in detail when the target is a Lie group of Hermitian type. Tight homomorphisms between Lie groups of Hermitian type give rise to tight totally geodesic maps of Hermitian symmetr…

A chord index homomorphism for knots in thickened surfaces is constructed.

Study classifies biharmonic and harmonic homomorphisms between specific Lie groups.

Study on Euler class and flux homomorphisms for non-orientable surfaces.

We examine functorial and homotopy properties of the exotic characteristic homomorphism in the category of Lie algebroids which was lastly obtained by the authors in [4]. This homomorphism depends on a triple (A,B,$\nabla$) where B $\subset$ A are regular Lie algebroids, both over the same regular foliated manifold (M,…

We define an infinite family of linearly independent, integer-valued smooth concordance homomorphisms. Our homomorphisms are explicitly computable and rely on local equivalence classes of knot Floer complexes over the ring $\mathbb{F}[U, V]/(UV=0)$. We compare our invariants to other concordance homomorphisms coming fr…

A new homomorphism connects group actions on circles to Euler classes.

We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improve…